Electronic Properties of Luttinger Liquid with Electron-Phonon Interaction by Alexey Galda A thesis submitted to The University of Birmingham for the degree of DOCTOR OF PHILOSOPHY Theoretical Physics Group School of Physics and Astronomy College of Engineering and Physical Sciences The University of Birmingham September 2012 University of Birmingham Research Archive e-theses repository This unpublished thesis/dissertation is copyright of the author and/or third parties. The intellectual property rights of the author or third parties in respect of this work are as defined by The Copyright Designs and Patents Act 1988 or as modified by any successor legislation. Any use made of information contained in this thesis/dissertation must be in accordance with that legislation and must be properly acknowledged. Further distribution or reproduction in any format is prohibited without the permission of the copyright holder. Abstract This thesis addresses a theoretical study of the problem of a single impurity embedded in a one-dimensional system of interacting electrons in presence of electron-phonon coupling. First we consider a system with a featureless point-like potential impurity, followed by the case of a resonant level hybridised with a Luttinger Liquid. The stress is made on a more fundamental problem of a featureless scatterer, for which two opposite limits in the impurity strength are considered: a weak scatterer and a weak link. We have found that, regardless of the transmission properties of phonons through the impurity, the scaling dimensions of the conductance in these limits obey the duality condition, ∆WS ∆WL = 1, known for the Luttinger Liquid in the absence of phonons. However, in the case when the strength of phonon scattering is correlated with electron scattering by the impurity, we find a nontrivial phase diagram with up to three fixed points and a possibility of a metal-insulator transition. We also consider the case of a weakly interacting electron-phonon system in the presence of a single impurity of an arbitrary scattering potential. In the problem of a resonant level attached to the Luttinger Liquid we show that the electron-phonon coupling significantly modifies the effective energy-dependent width of the resonant level in two different geometries, corresponding to the resonant and anti-resonant transmission in the Fermi gas. ACKNOWLEDGEMENTS This work would not have been possible without the tremendous support from my supervisors, Prof. Igor V. Lerner and Dr. Igor V. Yurkevich. They have always been truly helpful and understanding, not only in scientific matters but also in other spheres of life. This thesis reflects the results of the work I have done over the course of my PhD. I am grateful for all the time that Igors devoted to working on this project and to me personally. A special thank you goes to everyone who supported my desire to work in the field of theoretical physics and helped me to advance at the earliest stages of my career, especially to Dr. Sergey Artemenko and Dr. Alex Savchenko. A part of this work was done in collaboration with Dr. Oleg Yevtushenko, whom I thank for many valuable discussions. My PhD programme was financially supported by the scholarships provided by the Midlands Physics Alliance Graduate School and the Overseas Research Student Association. I acknowledge the hospitality of the Faculty of Physics, Ludwig-Maximilians-University in Munich, where part of the work was done, as well as the Abdus Salam International Center for Theoretical Physics where the final chapters of this thesis were written. I would like to thank all my friends and colleagues from the University of Birmingham who made my four years there extremely enjoyable. And, last but no least, I thank my parents for their constant support and encouragement. This thesis is dedicated to my parents. PUBLICATIONS • Impurity Scattering in Luttinger Liquid with Electron-Phonon Coupling Alexey Galda, Igor V. Yurkevich, Igor V. Lerner Physical Review B 83, 041106(R) (2011), Editor's Suggestion • Effect of electron-phonon coupling on transmission through Luttinger Liquid hybridized with resonant level Alexey Galda, Igor V. Yurkevich, Igor V. Lerner Europhysics Letters 93, 17009 (2011) • Duality of Weak and Strong Scatterer in Luttinger liquid Coupled to Massless Bosons Igor V. Yurkevich, Alexey Galda, Oleg M. Yevtushenko, and Igor V. Lerner to be published CONTENTS 1 Introduction 1 1.1 Motivation . .1 1.2 Outline of the thesis . .3 1.3 Fermi Liquid and Luttinger Liquid . .6 2 Techniques used 12 2.1 Keldysh technique . 12 2.2 Functional bosonisation . 15 3 Model Under Consideration 23 3.1 Pure Luttinger Liquid . 23 3.2 Impurity in a Luttinger Liquid . 26 3.3 Electron-phonon liquid . 30 3.4 Problem formulations . 36 4 Phonon-transparent impurity 38 4.1 Weak Scatterer . 39 4.1.1 Functional bosonisation solution . 40 4.1.2 Brute force method . 42 4.2 Weak Link . 44 4.2.1 Brute force method . 45 4.3 Discussion . 48 5 Phonon-nontransparent impurity 52 5.1 Weak Scatterer . 53 5.1.1 Brute force method . 53 5.2 Weak Link . 55 5.2.1 Functional bosonisation solution . 55 5.2.2 Brute force method . 57 6 Correlated electron and phonon scattering 59 7 Potential scattering in weakly interacting electron-phonon liquid 64 7.1 Corrections in the phononsless problem . 65 7.2 Corrections due to phonons . 68 7.2.1 Correction to the transmission amplitude . 71 7.2.2 Correction to the reflection amplitude . 73 7.3 Comparison with the Weak Scatterer and Weak Link results . 74 8 Resonant impurity 78 8.1 Resonant level . 80 8.1.1 Transmission coefficient . 86 8.1.2 Resonant conductance . 88 8.2 Anti-resonant level . 89 8.2.1 Transmission coefficient . 91 8.2.2 Resonant conductance . 92 9 Conclusions 93 Appendix A Polarisation operator 98 Appendix B Jacobian of the gauge transformation 100 Appendix C Inversion 104 Appendix D Renormalisation group 105 Appendix E `Unfolding' of the weak link geometry 109 Appendix F Zero-temperature Green functions 111 List of References 113 LIST OF FIGURES 1.1 Zero-temperature particle momentum distribution function in a) Fermi Liquid, b) Luttinger Liquid. Dashed line corresponds to the non-interacting Fermi gas.9 1.2 Particle-hole excitation spectrum in a) Fermi Liquid, b) Luttinger Liquid. 10 2.1 Closed Keldysh contour. Any time on the first part of the contour, C1, going in the positive direction, is assumed to precede the time residing on the second part of the contour, C2, going in the negative direction. 13 2.2 Linearisation of the electron spectrum in one dimension. 16 4.1 Renormalisation group flows of uncorrelated electron and phonon scattering from impurity. A transition between an insulator (G ! 0) and ideal metal (G ! e2=h) occurs at α∗, which depends on phonon parameters. 50 6.1 Bending of a carbon nanotube at angles between 0 and 90 [24]. A kink in lattice structure is formed at high bending angles, which can reflect electrons and phonons. 60 6.2 Renormalisation group flows for different α when electron and phonon scat- tering from the impurity are correlated. We assume that both are almost fully transmitted through the impurity in the weak scattering limit and almost fully backscattered in the weak link one; in this case ∆WS and ∆WL equal to 1 at different values of α, say α1;2. For α1 < α < minfα2; 1g the insulator (G = 0) and metal (G = e2=h) fixed points are both stable so that an unstable fixed point should exist at finite G < e2=h corresponding to a metal-insulator tran- sition at some intermediate value of the bare backscattering. Depending on a value of β ≡ v=c, such a line might (a) end at G = 0 dividing the phase dia- gram in the regions of insulator, α ≤ α1, metal, α2 ≤ α < 1, or metal-insulator transition, α1 < α < α2; or (b) end at the Wentzel-Bardeen instability line, α = 1, in which case the purely metallic region is absent. 63 7.1 Possible electron-electron scattering sectors (spin-irrelevant): g4 corresponds to the forward scattering (q ∼ 0) of electrons of the same chirality, g2 - for- ward scattering (q ∼ 0) of electrons of opposite chirality, g1 - electron-electron backscattering (q ∼ 2kF ). Right(left) - moving electrons are denoted by cor- responding indices. 66 7.2 First order corrections to the transmission amplitude. 76 7.3 First order corrections to the reflection amplitude. 77 8.1 Two geometries under consideration: resonant barrier geometry (top) and side-attached geometry (bottom). 79 ∆WL2 −1 8.2 Resonant-barrier geometry: Γoff / j"=E0j with the value of ∆WL2 de- pending on the electron-phonon coupling strength. Γdiag = Γoff at " & "0 and saturates at "0 (dashed line) for " . "0...................... 87 8.3 The renormalisation group scaling dimensions of the effective resonance width for the resonant barrier geometry, ∆WL2 − 1, and the side-attached geometry, ∗ ∆0 ≡ (∆WS1 + ∆WL2 )=2. Here K ≡ minf1; v=(vF αph)g is the boundary of the applicability region: we demand α < 1 to stay away from the Wentzel{Bardeen instability [49]. 89 ∆0−1 8.4 Side-attached geometry: Γdiag = Γoff / j"=E0j for " & "0;Γdiag saturates ∆WS1 −1 and Γoff / j"="0j for " . "0. The sign of (∆WS1 − 1) depends on the electron-phonon coupling strength, while ∆0 > 1. 91 E.1 `Unfolding' of the weak link geometry. 110 CHAPTER 1 INTRODUCTION 1.1 Motivation Continuous progress in semiconductor technology over the past decades has stimulated great interest in studying transport properties of low-dimensional electron systems. Recent ad- vances in the fabrication technology of two-dimensional electron gases have led to the cre- ation of effectively one-dimensional systems in which electrons are confined within a narrow channel, called quantum wires.